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NASA-CR-191716
DEPARTMENT OF MATHEMATICAL SCIENCES
COI.IFGE OF SCIENCES
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
THEORETICAL STUDIES OF A MOLECULARBEAM GENERATOR
By
John H. Heinbockel, Principal Investigator
Progress Report
For the period May 16, 1992 to November 15, 1992
Prepared for
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23681-2199
Under
Research Grant NAG-l-1424
Dr. Sang H. Choi, Technical Monitor
SSD-High Energy Science Branch
(NASA-CR-191716) THEORETICAL
STUDIES OF A MOLECULAR BEAM
GENERATOR Progress Report, 16 May -15 Nov. 1992 (Old Oominion Univ.]
52 p
G3/72
January 1993
// ///
:c) r- j
N93-169_5
Unclas
0139659
https://ntrs.nasa.gov/search.jsp?R=19930007756 2020-05-25T18:37:35+00:00Z
DEPARTMENT OF MATHEMATICAL SCIENCES
COLLEGE OF SCIENCES
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
THEORETICAL STUDIES OF A MOLECULAR
BEAM GENERATOR
By
John H. Heinbockel, Principal Investigator
Progress Report
For the period May 16, 1992 to November 15, 1992
Prepared for
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23681-2199
Under
Research Grant NAG-l-1424
Dr. Sang H. Choi, Technical Monitor
SSD-High Energy Science Branch
Submitted by the
Old Dominion University Research FoundationP.O. Box 6369
Norfolk, Virginia 23508-0369
January 1993
PROGRESS REPORT
RESEARCH GRANT NAG-l-1424
ODU RESEARCH FOUNDATION GRANT NUMBER 126611
THEORETICAL STUDIES OF A MOLECULAR BEAM GENERATOR
MOLECULAR BEAM GENERATOR MODEL
The following is a proposed baseline model that is being develope for the simulation of
hydrodynamic generator, which can be converted at a later date to a magnetohydrodynamic
MHD thruster by adding the necessary electric and magnetic fields. The following development
will include the electric and magnetic terms, however, the initial computer program will not
include these terms. The analysis that follows is for a one species, single temperature model
constructed over the domain D defined by the region enclosed by ABCDEF illustrated in the
figure 1.
Figure 1. Geometry of thruster MHD model.
CONTINUITY EQUATION
The continuity equation expresses conservation of mass and is given by
cOpo-7+ v .(pff) = o (1)
1
where p = p(r, O, z, t) is the density of the gas, and I_ = Vr _r + VO _o + Vz _z is the velocity. In
cylindrical coordinates the equation (1) has the form
Op io(,-pV,)iO(pvo) O(pv )-=+ + + -o. (2)r Or r (9t9 OzOt
CONSERVATION OF MOMENTUM
The equation for conservation of linear momentum is given by
np---_ + p(V. V)V = ___ ffi (3)
i=1
where _n=l ffi represents a summation of body forces per unit volume acting upon a control
volume within the domain D. We consider initially the pressure force
F1 = -VP (4)
where pressure and density are related by the equation of state gas law P = p*RT, Where p* is
the density in mole/m 3. i.e. p*W = p where W is the molecular weight in kg/mole. The force
due to viscosity is
if2 = rl { v2_ + V(V. ¢)} - _V(71V. I7)+ 2(Vr/- V)V + Vr/× (V × V) (5)
where
r1=1"2510-19 5 ( Mm-k_ BT)8niA
is the plasma viscosity, with
A = 2(log(1 + a 2)
1/2 (2kB_ T)2=C1T 5/2
o_2
I+o12 )
(6)
a constant which depends upon the ionization factor a, (a = 1 for a fully ionized gas). The
additional constants are M the ion mass (Kg), m the electron mass, e the electron charge, k B
Boltzmann's constant, T is the absolute temperature, and ni = 1 for a singly ionized plasma.
For oL = 0, we employ an empirical curve fit for the viscosity as a function of temperature. The
magnetic force is given by
g3 =fxB (7)
22
where f is the current density. The gravitational force is given by
if4 = P_'.
The electric force is given by
All additional forces are represented by
and are neglected for the present.
12
i'6
CONSERVATION OF ENERGY
Representing the internal energy by u = CpT where Cp is the specific heat at constant pressure
and T is the absolute temperature, the energy equation can be written in the form of an energy
b alatlce a_s
where both
temperature T.
relation
O(CpT) "P 0t + p(f" V)(CpT) = V(KTVT) + _ ¢i (8)
i=1
the specific heat Cp and thermal conductivity KT are treated as functions of
The thermal conductivity K T of the medium is given by the Spitzer-Harris
4.4 10 -l° T 5/2KT = (9)
23-1og[ l'22103nl/2]Ta/2
and n is the plasma number density in particles/m 3. In addition to the heat loss term the right
hand side of equation (8) contains the terms
¢1 =(_+ f x B). Y-0f" (I0)
which represents joule heating,
+ + \-o-7 + 0r )
Iov + A\-6V + -7 + -_z ]
-87/]
(0v0 v0)+ k, Or r 2} (11)
3
which represents viscous dissipation. Here r/and A are viscosity coefficients satisfying A + _q = 0.
In addition there is the radiation loss term. Various forms of the radiation term exists in the
literature. As a first approximatrion we take the radiation loss term from reference 11 which can
be expressed,4
¢3 = wR v(_T 4)
where )_R is the Rosseland mean free path (,k R = 1/a R where _R is the Rosseland absorption
coefficient (cm-1)), and a is the Stephan Boltzman constant. The remaining terms )--_n=4 ¢i
represents additional energy considerations which are initially neglected.
MAXWELL'S EQUATIONS
Maxwell's equations in the MKS Rational system of units can be expressed
Gauss's law for magnetism
Gauss's law for electricty (Coulomb's law)
Ampere's law
Faraday's law
v.g=o (12)
v.b=p_ (13)
VxH=f+ 0/_- (14)Ot
Vxg= ogat (15)
CONSTITUTIVE EQUATIONS
Assuming an isotropic, homogeneous medium we adopt the constitutive equations
/)=e/_ and /_=#H.
OHM'S LAW
Ohm's law is written in the form
f_ -.
..f= o (_i+ 17x g)- _.(J x .g) (16)
where f is the current density, /_ is the electric field, /_ is the induced magnetic field and a is
the electric conductivity with milts of mho/m and f/is the Hall parameter given by (reference 1)
£l = 9.6(1016)(Ta/2B/Zn logA)
4 4
with Coulomb logarithm given by
logA _ 23 - log(1.22x 103nU2/T312)
with n the plasma number density.
ELECTROMAGNETIC FIELD EQUATIONS
Neglecting the displacement current modifies Ampere's law to
v×g=#Z (17)
Assumming that the charge density is constant implies the equation of continuity of charge
is V. f = 0. (Note that the divergence of equation (17) also gives this result.) From Jackson,
reference 10, along with neglecting the displacement current, it is appropriate to ignor Coulomb's
law as its effects are negligible. We thus obtain the electromagnetic field equations
v×g=j
Vx._- agat
(is)
Using equation (17) in Ohm's law we solve for/_ and write
= _v x g- 17x g + _'(v x fi) x g (19)
where/3 = _t/tzBa and a -- 1/ap. We substitute the results from equation (19) into Faraday's
law and write
o.d-.-_-.-= v x (av x/3)- v x (17x/3) + v x ,8(v x/_) x B (20)
and since/3 is a function of T we find
ag-Vx(aVx/3)-Vx(17xB)+V/_x(Vx/3) x/3+/3Vx(Vx/3) x/3 (21)
Ot
5
SUMMARY OF BASIC EQUATIONS USED FOR MODELING
Continuity
Moment um
Opo-_+ v(p¢) = o
P-k- + p(p v)_ = _._i=1
Energy
n
+ p('V. V)(CpT) = V(KTVT ) + _ ¢ii=1
Electromagnetic field equations
ogOt
--- = v x (_ x_)- _ x (_x _)+ v x Z(vx _)x
This produces a system of eight simultaneous partial differential equations in the eight
unknowns
Br, BO, Bz , p, Vr, Vo, Vz , T.
Throughout the calculations the following quantities can be generated in terms of the above
variables.
Y=ivx_#
._= o_,f- ff x g + #B(Zx g)(22)
SCALAR FORM OF FIELD EQUATIONS
Assuming symmetry with respect to the 0 variable, all derivatives with respect to 0 are
neglected. The following set of scalar equations then results
Continuity
op ia(_py_) o(pyz)--_-+ + -or Or Oz
6
Momentum
n
or, (v,°v" or, v_ =_(_,),P---_- + O t --'_'V -l- Vz Cgz r i=1
n
p---_- + p Vp + Vz -]- =i=1
n
P---_--}-P Vr-_r +Vz'-_-z J i=1
Energy
p (gp + Ta_6____ _T agp'_ a_T _-T =\ ux l
aT \\-bT) + \_--T) ]+KTI_2+-/a---/+-_ 2]
I'i
+ _,_ii=1
7
Electromagnetic field equations
OBr
&
02Bo OB_OBo (02Bo- _ B"-O_z2 + Oz Oz + B" k o-_z
+82_[ z 02 +B,.\ Or +OB 0 02Bo 02Bo _ OB o
Ot c_ Oz 2 oL 0r 2 r Or
___ OV_ OBo OV,.+ v_ + Bo--_-z + v,--gV + Bo-g_
+5 Bz Oz 2 OrOz + _ Oz
+ B, \ o-7_z o_2 . + _ \ Oz
+ -_z Bz Oz Or - Bo \ Or
OBz 02Br 02Bz _ OBr _ OBz
Ot - _ OrOz -_ + r Oz r Or
V, OBz OV, 1+ --_-r +Bz-_r +r(VrBz-VzBr)
02BO OBz OB 0 (02Bo+ fl Bzo---_z + Or Oz + Br\ Or 2 +-_
+_ Bz--0T+B" \ 0_ +
OBz OVr
Oz Oz
_+7 o--7- +-g7 \ o, +
02 Br 02 Bz OBr Br OVz_-gy + _ O-_z + V,-N-z + -_-z - y,
ol OB z OVo
+ -_Bo- Vo_ Bz O---;-OBr OVo
-- vo O--g--B, O--;-
OBz "_ 2Be OBe
Or )
0_ ]J+
r Oz
O_ [,., OB 0 (OBr OBz_]L_O-_z + B, Oz _ ]
OBr OVz
v_ 0---7-- B, 0--7-
o_ +--g# k-g-, + + r _j
These equations are subject to certain boundary and initial conditions which are now
discussed.
BOUNDARY AND INITIAL CONDITIONS
With reference to the figure 1, the line AF has the input conditions
P = P0 = constant
T = To = constant
V,. = Vo =0
Vz = Vo = constant
OB,. OBo OBz
Oz Oz Oz
8 8
Due to symmetry considerations the line AB has the center line boundary conditions
Op =0Or
OT_--'0Or
01I._-=0Or
01Io 0Or
OVz_'--0Or
OBr-0
Or
OBo = 0Or
OBz-0
Or
The far field conditions along the line BC are given by
°! = o oyz = oaz Oz
aT cOB,.=0 -0
Oz Oz
OVr OBo-0 _---0
Oz COz
011"o OBz-0 -0
Oz Oz
For the insulated boundary segment between ED we assign the boundary conditions
V,. = Vo = Vz = 0 no slip boundary condition
T = To = constant
0-2-0= 0Or
B,.=Bo=Bz=O
For the uninsulated boundary segments FE and DC we assign the conditions
Vr = VO = Vz = 0 no slip boundary condition
Jo = 0 which implies Oz
O(_Bo)Or
T = To = constant
0__pp=0Or
COB,. OBz
- Or and simultaneously
=0 and OB____O0=0Oz
9
These later boundary conditions upon B insures that the electric field J_ satisfies the condition
/_- i* = 0 everywhere on the nozzle boundary, where t' represents a unit tangent vector to an
arbitrary point on the nozzle boundary.
Initial conditions are assigned in order to avoid large initial transients in the numerical solution
because large changes can lead to numerical instabilities of the system of partial differential
equations. We therefore assign the following initial conditions at all interior grid points.
T = To = constant
Vr = Vo =0
Vz = VO = constant
Br, Be and Bz are assigned values such that V •/3 - 0
everywhere in the solution domain.
TRANSFORMATION OF COORDINATES
We make the change of variables
Z r
x b Y- f(z)
so that the domain of the nozzle 0 < r < f(z), 0 < z _ b transforms to the computation domain
0 < x < 1 and 0 _< y _< 1. The scalar form of the governing equations can then be written in the
following forms.
Continuity
Op i O(pV_) pv_ i o(pyz)o7 + SS) +yT(;)+i
yf'(z) O(pVz)
f(z) Oy
Momentum Equations
-0
OV_
Ot
Oy_Ot
Vr OVr (10Vr yf'(z) OVr'_--+ f(z) Oy + Vz b Ox f(z) _,]
Vr OVo (10Ve yft(z) OVo--+ f(z) Oy + Vz b Ox f(z) Oy ] +
Oyz y_
Ot f(z)
n
yf(z) P i=1
12
v vo _ !yf(z) P i=1
vf'(z)OV_) 1 _:(z) =i=1
OV_ { 10v_Oy + V_ _ b Ox
10 10
Electromagnetl c Field EquatiOnS
2
ez-component
OBz
Ot
1 02Bz
fX(z) Oy 2
ft(z) OBr 1 02Br yff(z) 02Br- o_ f2(z) Oy + bf(z) OxOy f2(z) Oy2
1 [_0B_ Vf'(z) OB_] 1+ yf(z) Ox f(z) _y J yf2(z)vz OBr Br OVz V_ OBz
+ +f(z) Oy f(z) Oy f(z) Oy
{ (-..ft(z)OB°oY bf(z) Oxoyl02B°+ fl ez "f2(z ) +
10Bz (_OB o yf'(z) OBo)+ f(z) ov ox f(z) ov
Oy )B: Ov,. GB,. V_B:
+_f(z) Oy yf(z) yf(z)
yf'(z) O2BO_
f(z) ov2 )
Br 02Bo Br OBo 1 0Br (10Bo Be )+ i2(z) Oy2 + y]2(z) oy + f(z) oy f(z) oy + y]-_
+v-]-_ b ox f(,) ov + vf2(,) ov
fl'(T) OT [ (lOBe yf'(z) OBo) ( 1 OBe Be )]+ f(z) Oy Bz b Ox f(z) Oy + Br f_z) Oy + yf(z----)
NUMERICAL SOLUTION
We are primarily interested in the steady state solutions and the time necessary to achieve
steady state. The above system of coupled nonlinear partial differential equations are simplified
by assuming symmetry with respect to the 0 variable. This enables us to set all derivatives
with respect to 0 equal to zero. Additional assumptions regarding magnitudes of force terms and
energy terms can be made by doing a dimensional analysis of the resulting system of equations. A
grid generation technique is used to alter the solution domain to a rectangle. Then the equations
and rectangular boundary can be scaled before any numerical solution techniques are applied.
The system of equations are then solved numerically using ADI (Alternating Direct Implicit)
techniques patterned after the Lax modification.
GRID GENERATION
Let r = f(z) denote the nozzle boundary for 0 < z < b and consider the mapping from
the (r, z) real coordinates to the (x,y) computational coordinates given by the transformation
equations
X -- --
br r (23)
y __ __ __
_._ f(z)
13
where rmax = f(z) denotes the nozzle boundary which changes with position.
converts the region
D = { r, z l O < z < b, O < r < rrnax }
This mapping
to the region D' of computational coordinates given by
D'={x, ylO<x<l , 0<y<l}.
To handle large gradients in any of the independent variables, the computational x, y domain is
partitioned into 6 regions as illustrated in the figure 2.
o /.o
Figure 2. Computational coordinates
The 6 regions are characterized by the selection of the Ax and Ay step sizes. In this way finer
grids can be specified near the boundaries and nozzle regions where large gradients can occur.
Observe that any partial differential equation of the form
. 02u .02u Ou OuO--U-uOt= Dl(r'z)_ _-2+ D2(r'z) o--_zz + D3(r'z)--_z2 + D4(r'z)-Orrr + D5(r'z)-_z + D6 (24)
1414
where u = u(r, z, t) and D6 = D6(r, z,t, u,...) can be converted to computational coordinates
x, y by using the chain rule for partial derivatives. These changes can be represented
Ou Ou Oz Ou Oy
Or Ox. Or Oy Or
Ou Ou Ox Ou Oy
Oz Ox Oz Oy Oz
and02 u Ou 02z Oz
Or---'_ -- COx cot 2 + -_r
COu CO2y COy
+ COyCOt2 + "_r
02 u COu02 x COx
COz---y = COxOz 2 + -_z
COu CO2y COy
+ CO_COz---_+CO2u COu CO2z
COrCOz COzCOrCOz
CO2u COx CO2u Oy'
COx2 COr+ COzCOyOr
CO2u COx CO2u COy
COyCOx Or + COy2 COt
"CO2u COx CO2u COy
Ox 2 COz COzCOyCOz
CO2u cox 02u COyCOyCOxCOz -4- COy2 CO'--'z
az [co2ucox a2u ay ]+ _ Lcox2 coz+ coxcoy J
CO2u COY]+ COyCOrCOz -_r [COyCOzCOz + COy2
Then the partial differential equation (24) can then be written in the x, y computational
coordinates
or
COu -* CO2u * CO2u D* c32u , COu COu_-=DI_x2 +D20-_y+ 3-_y2 + D4"_x + D_-_y + D;
(25)
15
where
D_ = D_(x,y)= Dl(xr) 2 + D2xrXz + D3(xz) 2
D_ = D_(x,y) = 2Dlxryr + D2(xryz + yrxz) + 2D3xzyz
D; = Dl(Yr) 2 + D2YrYz + D3y2z
D_ = D_(x,y) = DlXrr + D2xrz + D3xzz + D4xr + D5xz
D_ = D_(x, y) = DlYrr + D2Yrz + D3yzz + D4yr + D5yz
D_ = D_(x,y)= D6(yf(x),x,t,u,...)
and Di = Di(r,z) = Di(yf(x),x) for i= 1,...,5 and
1
xr-0, yr-1/f(z), Xz--_, yz=-yf'(z)/f(z)
(f"(z)zr_ = _= = _zz = w_ = o, w= = -f'(z)/(f(z)) 2, v,== -v _, f(z)
Then the above coefficients reduce to
D;(x,y) =
D2
D_(x,y)- bf(z)
D1
D_(x,y)- (f(z)) 2
D_(x,y) = 95b
D_(x,y) = -D2--
S'(z) y2 (s'(z)y-2 (f-_2 + D3 \ f(z) )
f'(z) ( f"(z) ( ft(z) _ 2_ D4 yD5f'(z)(f(z)) 2 yD3 _ _ 2 t,,_] # / + S(z-_-)- f(z)
D_(x,y) = D6(yf(z),x,t,u,...)
where
Di = Di(r,z)= Di(yf(z),x), for i= 1,2,3,4,5.
ADI NUMERICAL METHOD
The following description of the ADI numerical method is for uniform Ax and Ay grids and
of course has to be modified for unequal x and y spacing. In step 1 of the ADI numerical method
the interior points to the region D I of the computational domain are labeled from left to right as
illustrated in the figure 3. Assume that the system of partial differential equations to be solved
16 16
have all beennormalized. Partition the segmentfrom x = 0 to x - 1 into segments with spacing
Ax = 1/m2 so that the ith node is iAx and the right boundary is m2Ax, with 0 _ i _ m2.
Similarly, partition the segment from y = 0 to y = 1 into segments with spacing Ay = 1/ml so
that the jth node is jay and the top boundary is relay with 0 < j _< ml, where i,j, ml and
m2 axe integers. The interior points to this grid are then labeled as illustrated in the figure 3.
Let un be associated with the (i,j)th node point, where
n = (m2 - 1)(ml - 1 - j) + i ml and m2 are fixed.
Conversely, given the Un point, we can determine its position i, j from the relations
j -- ml - 1 - Int[(n - 1)/(m 2 - 1)]
i = n - (m2 - 1)(rnl - 1 - j)
where Int[x] is the integer part of x.
Ul u2 u3 u4
Um2 um2+l Um2+2 Um2+3
• • • ,
• • • Urn2-1
• • • U2rn_ --2
Un
• "" u(m2_1)(ml_1)
-(_. 9--
m
Figure 3. Step 1 labeling of interior points to domain D
17
Letting
u(iAz,jAU, nat) = u-"-I,J
and dropping the star notation, we then replace all partial differential equations of the form of
equation (31) by difference equations having the form
u n+l - u.'3.l_J l,J
At
/ n+l _ 2un+l un+l \
=D1 lUi+l,j (Ax)2i'j "+- i-l,j)
( u n -- un _ u n )+ D2 i+l,j+l i+l,j-1 i-l,j+l nt" Ui-l,J -1
4AxAy
"t- D3 i,j+l - i,j-1 [Ui+l,j___ i-l,j(Ay)2 + D4 \ 2Ax
+ D5 k, 2Ay + D6
which can be rearranged to the form
= 1 (Ay) 2 / 'J + (Ay) 2 q- 2Ay / i,j+l h- \_
D1 At _ un+ l(Az)2 ] i-l,j
D5At "_ un. .// 2,3-1
U n _ U nD2At ( i+l,j+l un i-l,j-1) + DOAt+ 4AxAy - i+l,j-1 Un-l,j+l nt-
Evaluating this equation at each of the interior node points gives rise to a system of
(m2 - 1)(ml - 1) implicit linear equations which are then solved by row reduction methods.
The second step of the ADI method relabels the interior points of the computational domain
from top to bottom as illustrated in the figure 4.
For this labeling we can let Un denote the point associated with the (i,j)th node where
n = (ml - 1)(i- 1)+ -j.
Conversely, given Un we can solve for i and j from the relations
i= 1 + Int[(n- 1)/(ml- 1)]
j=(ml-1)(i-1)+ml-n
1818
Ul Ural .........
u2 Um_ + l .........
u3 Urea+2 .........
".. : :U4 Uml+3
:...... Un
Urn1 -1 U2m1-2 .........
mmm
-4i. j}
m
Figure 4. Step 2 labeling of interior points to domain D'
For step 2, all the partial differential equations of the form of equation (31) are replaced by the
difference equations having the form
ttn+l -- un" "i,jz,a = D1 (un+l'j - 2u_'j + un-l'j)
n _ u nq_ D2 Ui+l,j+l - un+l,j-1 i-l,j+l q- Ui-l,j-1
)4AxAy
n+l -- 2u 'n.+l -t- ui,j_ 1 n n+ D3 ui'j+l t,a
-(_y)2 + 94 \ 2Ax
n+l n+l )
Ui,j+ 1 -- ui,j_ 1
+ D5 -2Xy + D6
19
which canbe rearranged to the form
1+ T£-_] i,j \(-h-_u)2 + 2a---T -i,_+1+ \ 2a_ (ay)2/ _-1
D2At (uin+l,j+l uni-I-1j-1 Un+ 4AxAy - - i-l,j+l + u,-1,j-1/ + De At
Applying this difference equation to each interior node point results in an implicit system of
(ml - 1)(m2 - 1) simultaneous linear equations which must be solved for the values of u at the
node points.
One can see that the finer the interior grid there results a larger system of linear equations
to be solved. Also the results from the ADI numerical method are more accurate on the even
numbered time steps. Additional complications results when employing the unequal step size
approximations illustrated in the figure 5. The unequal step sizes are necessary to handle large
gradients occurring in any of the dependent variables. The computational region is therefore
divided into 6 regions as illustrated in the figure 5. The density of node points can be changed
in each region by selecting different step sizes in the computational coordinates.
1
4
2
5
3
6
Figure 5. Unequal x and y spacing.
2O2O
In the case of unequal grid spacing we employ the difference approximations.
0_ _ hxl + z_x2 (u(x
Ou 1 [ Ay2 , ,
02u 2 [u(x + Axl,y)-- u(x,y)
Ox-_ "_ Axl + Ax2 [ Axl
02u 2 [u(z,y + Ayl)-- u(x,y)
LOy 2 Ayl + Ay2 Ayl
02u u(x + Axl,y + Ayl) -- u(x + Axl,y-- Ay2)
OxOy (Axl + Ax2)(Ayl + Ay2)
u(x -- Ax2, y -- Ay2) - u(x -- Ax2,y + Ayl)+
(Axl + Ax2)(Ayl + Ay2)
AXl / / ]+ a_l, v)- u(_, y))- X_2_u_x- ax2, v)- u(_, v))d
AVl(u(_, ]V+ a_l) - _(_,y))- _ V- aV2)- _(_, y))
_(_ - Ax2,V)- _(_, v))+
Ax2
u(x,y -- ay2) -- u(x, y))+
Ay2
SPECIAL CASE-ELECTRIC FIELD IN VACUUM
In a vacuum we solve02 ¢ 1 0¢ 02 ¢
v2¢ - _ + T_ + b_z2= 0
over the domain 0 _< r < f(z) and 0 < z < 1. Using the transformation equations
(26)
z=x r=yf(z)
where
f(z) = .2 + x tan(Sir/IS0)
is used to describe a straight line nozzle boundary. The equation (26) then transforms to
02¢ ,02¢ ,02¢ 0¢0_2 + 4_' v_0--V_y+ b(x'v_b-_2+ 4_'_) N = 0
over the domain 0 _< x _< 1,0 < y < 1 where
_2yf'(z)a(x,y) = f(z)
1 2b(x,y) = f2-(z ) + k, -f-(_]
1 (f"(z)_(x,v)- vf2(z) v \ f(z)
(27)
21
• o° . .
F/d_"_'E/v" POTENTIAL FIELD IN VACUUM
5O 5O
10 1o
22
MAGNITUDE OF ELECTRIC FIELD IN VACUUM
_5000
4000
3000
tO00
30'
2O
10 10
23
FI'_JJfE P.POTENTIAL FIELD IN VACUUM
_< o.
0 -200
-400
5O
3O
10 10
24
MAGNITUDE OF ELECTRIC FIELD IN VACUUM
;000
2000
1000
3O
10 10
2O
- -i
25
Using the notation ui, j = u(iAx,jAy) and the difference approximations
Uxx
Uyy
Uy
Ui+l,j -- 2ui,j + Ui-l,j
h 2
Ui+l,j+l - Ui+l,j-1 - Ui-l,j+l + Ui-l,j-1
4h 2
ui,j+ 1 -- 2ui, j + ui,j-1
2h
and defining
d(x,y) = _2(1 + b(x,y))
the equation (26) reduces to the difference equation
ui'j = _ijl f Ui+l,j h2+ Ui-l,j + _aij (Ui+l,j+ 1 _ Ui+l,j-1 -- Ui-l,j+l + Ui-l,j-1)
bij cij(Ui+l Ui-l,j) }h"-_ (ui,j+l + ui,j-1) + _ 'J -
This difference equation is subject to the boundary conditions
Ui,jmaz = assigned potential value
uo, j = u2, j
Uimax,j = Uimaz-2,j
ui, 0 = ui, 2
which represent zero derivative boundary conditions along the other three sides. The figures 6,7,8
and 9 illustrate the potential function for two different nozzle configurations where the cathode
is assigned a value of -500 volts and the anode(s) is assigned a value of +500 volts.
FLOW AND HEAT TRANSFER THROUGH A POROUS MEDIA
In the figure 10 a porous material is heated with 40kw of power from a solar simulator. We
assume that the solid porous material is heated to a uniform temperature Ts and that a gas flows
through the porous material and is heated.
22
26
Q tad cooling
Q Q conv
>
co_oling
Figure 10. Heat transfer through porous material.
27
In the following discussions we use the notations:
¢=porosity 0<¢<1
Tg, Ts = Temperature of gas and solid (g)
pg, ps = Density of gas and solid (gm/cm 3)
Kg_ Ks =
Ke =
tt_
O_g
L=
Thermal conductivity of gas and solid (cal/scm 2 K/cm)
(1 - ¢)gs + CKg = Effective thermal conductivity
Velocity of gas (cm/sec)
Kg/Cpgpg = Thermal diffusivity of gas (cm2/sec)
Thickness of porous material (cm)
(dimensionless)
Radius of disk (cm)
Heat transfer coefficient (cal/s cm 2 K)
Input power (Kw)
Surface area of disk (cm 2)
Specific heat of gas and solid (cal/gmK)
Ratio of surface area to volume of porous media
Dimensionless temperature ratio
Dimensionless temperature ratio
uLPe = -- = Peclet number
O_g
h=
Q0 =
A=
Cpg, Cps =
Rs --
U= Ts/Tgo =
Y = Tg/Tgo =
tur = -- = Dimensionless time
L
X x= - = Dimensionless distanceL
(cm2/cm a)
Following reference 1, the basic equations describing the heat transfer to a gas moving through
a porous media are given by
( OTg OTg O2T,pgcp_ \ Wi- + U-gT ] = _'9
hR(Tg - Ts) (28)
O=TsOTs _ Ks hR(Ts - Tg) (29)p_Cp_ & Oz2
for0<x <L andt >0.
24
28
and
The equations (1) and (2) are subject to the boundary conditions
gs OTs 239Q0 ea( T 4 4-ffz'z ]z=O - -A - T;O) - pgCpgu(Ts - Tgo )
aT, = h(T, - T_ )-Ks
where all terms have been scaled to the units of cal/s cm 2 and
(30)
(31)
a = 5.67(10-12)(.239) cal/s cm 2 K 4
is the Stephan-Boltzman constant, e is the emissivity.
conditions
aTg = hR(T_ - T_)pgCpg_ -_ z=oaT, aTgozl.=L=0 and I =L
We further assume that the initial conditions are
In addition we have the boundary
=0
(32)
(33)
Ts=Tg=Tgo. (34)
Introducing the dimensionless variables U, V, X, r, the above equations can be written
OU / 1 02U
a--_ = Ao _ r_ OR 2
avov (_a2v0----_+ a--Z = A1 OR 2
OU
lR=,
10U
+ r_R OR
+
-hr
= -kT(u- 10
10V
r20R OR
1 02U_+ L2 _] - Bo(V - V)
1 02V)+ L2OZ 2 -BI(V-U)
whereL 2 hRsL 2 KsL 2 hRsL 2
A1 -- p--_ B1 - PeKg Ao = agpsCpsPe Bo = agpsCpsPe
The equations (35) and (36) axe subject to the boundary conditions
OU= A3 - B3(U 4 - V 4) - C3(U - V)
OZ z=o
OV
D-Z']Z=O = B4(U --V)
OU OV
oz Iz=,=o, =o
(35)
(36)
(37)
(38)
(39)
(40)
29
where
eaT_oL KoPe hRsL 2A3-- 239 .roR/LO( ) , B3 = --, C3-- _, B4 = (41)
AKeTgo Ke Ke ag Pe pgCpg
The initial conditions are
U(O,t)- 1 and V(O,t)= 1 (42)
We divide the intervals 0 _< Z < 1 and 0 < R < 1 into N parts with step sizes
AZ = AR = 1IN. We desire to represent the temperature of the gas and solid at the positions
R = iAR and Z - jZXZ for the time r = nat which is based upon the given temperatures
at time r. Let U(iAR, jAZ, nAT)= U .n. and V(iAR, jAZ, nAr)= V..n. and use difference2,? t,3
approximations to write the above equations as difference equations. We use the ADI (Alternating
Direction Implicit) method to solve the above system of coupled partial differential equations.
Material Properties
From reference 15, we obtained the following empirical data for Hafnium carbide.
Temperature Thermal Conductivity of Hfc
deg K W/cm K
560
800
II00
2000
2500
3000
0.09
0.12
0.13
0.15
0.25
0.29
The best fit second degree polynomial to the above data is given by
Ks(T) -- .0361887 + 1.03093(10 -4) T - 1.2077(10 -8) T 2
26
3O
Temperaturedeg K
60
95
300
600
1000
3000
Specific Heat of Hfc
cal/g K
0.011
0.020
0.045
0.065
0.065
0.065
Table lookup will be used to fit the Specific Heat data.
Temperature Emittance
deg K
300
1100
1900
2500
2900
1.00
0.98
0.90
0.70
0.62
The above data is represented by the approximating function
e(T) = .8- .2 tanh((T- 2100)/1000).
We use the above data to construct empirical relations to represent the constants in the above
system of coupled partial differential equations.The Appendix A contains graphical output from
the computer analysis of the heat transfer in a porous media.
31
REFERENCES
1. M.R. LaPointe, "Numerical Simulation of Self-Field MPD Thrusters",
AIAA/SAE/ASME , 27th Joint Propulsion Conference, June 24-26,
1091/Sacramento, CA, AIAA-91-2341.
2. E.H. Niewood, M. Martinez-Sanchez, "A Two-Dimensional Model of an
MPD Thruster", AIAA/SAE/ASME , 27th Joint Propulsion Conference,
June 24-26, 1991, Sacramento, CA, AIAA-91-2344.
3. E.J. Sheppard, J.M.B. Chantly , M. Martinez-Sanchez, "Local Analyses in
MPD Thrusters: Diffusion Reaction and Magnetodynamics Regimes",
AIAA/SAE/ASME , 27th Joint Propulsion Conference,
June 24-26, 1991, Sacramento, CA, AIAA-91-2589.
4. M. Auweter-Kurtz, S.F. Glaser, H.L. Kurtz, H.O. Schrade, P.C. Glaser,
"An Improved Code for Nozzle Type Steady State MPD Thrusters",
IEPC 88-040, Oct. 1988.
5. R.D. Gill, Editor, Plasma Physics and Nuclear Fusion Research,
Academic Press, 1981.
6. E.H. Holt, R.E. Haskell, Foundations of Plasma Dynamics,
Macmillan Co., N.Y., 1965.
7. J.A. Bittencourt, Foundations of Plasma Physics, Pergamon Press, 1986.
8. K.R. Cramer, S. Pai, Magnetofluid Dynamics for Engineers
andApplied Physicists, McGraw Hill Book Co., 1973.
9. A.B. Cambel, Plasma Physics and Magnetofluid-Mechanics,
McGraw Hill Book Co., 1963.
10. Jackson, J.D. Classical Electrodynamics, Second Edition,
John Wiley and Sons, Inc., 1975.
11. E. S. Oran, J. P. Boris, Numerical Simulation of Reactive Flow, Elsevier Science Pub-
lishing Co., N.Y., 1987.
28
32
REFERENCES CONTINUED
12.
13.
14.
15.
E.T. Mahefkey, J.D. Pinson, J.N. Crisp, Analysis of Transient Heat Transfer in Porous Solids
with Application toward Phase Separation Processes., ASME Paper 77-WA/HT-13, November
1977.
A. Hadim, L.C. Burmeister, Onset of Convection in Porous Medium with Internal Heat
Generation and Downward Flow., J. Thermophysics, Vol.2, No.4, October 1988.
L. Carlomusto, A. Pianse, L.M. deSocio, P. Kotsiopoulous, T. Calderon, Temperature
Distribution in a Porous Slab with Random Thermophysical Characteristics, Convective Heat
and Mas_ Transfer in Porous Media, NATO ASI series E, Kluwer Academic Publishers,
August 1990.
CINDUS, Purdue University, 2595 Yeager Road, West Lafayette, IN, 47906-1398
33
APPENDIX A
GRAPHICAL REPRESENTATION OF RESULTS
Solution of the coupled equations describing flow of gas through a porous media. The
input power in Kw is assumed to be a cosine curve of the form
7rR
Q(R) = 40. cos(-_-).
Solution of coupled equations is by ADI (Alternating Direction Implicit) technique.
All dimensions have been normalized. The radial and axial directions range from 0 to 1.
The temperatures of the gas (V) and solid (U) have been normalized by the equations
U = TS/TGO V = TG/TGO
where TGO = 300 K.
The figures A1 throug A16 illustrate the temperature change for the gas and solid as
a function of the normalized time
_U
L
where t is real time, u is velocity, and L is length in the axial direction.
34
!
,¢¢
Figure A1. Normalized Gas Temperature at T = 1.0
35
/
e'e
,'V
Figure A2. Normalized Gas Temperature at T = 2.0
36
/v
7-E-/V/_E-RA 7-UA_/--
Figure A3. Normalized Gas Temperature at T = 3.0
37
@
@
@.@
.¢
7-EIV_E-f_A 7-U_E/V
Figure A4. Normalized Gas Temperature at r = 4.0
38
e7-_---/V/_ERA 7UR/-
Figure A5. Normalized Gas Temperature at T = 5.0
39
AY
Figure A6. Normalized Gas Temperature at r = 10.0
4O
e'e
.¢
CD /L_
7-E/V/_ERA 7UAP±AJ
Figure AT. Normalized Gas Temperature at T = 20.0
41
0
Figure A8. Normalized Gas Temperature at T = 30.0
42
q
/u
7-E/V_E/qA 7-UAP/--/v
Figure A9. Normalized Solid Temperature at T = 1.0
43
q
/U
Figure A10. Normalized Solid Temperature at r = 2.0
44
Figure All. Normalized Solid Temperature at r = 3.0
45
A2
Figure A12. Normalized Solid Temperature at T = 4.0
46
q
.¢¢
/U
Figure A13. Normalized Solid Temperature at T = 5.0
47
@
Figure A14. Normalized Solid Temperature at r = 10.0
48
q
e"0
7-E-/V_E-_A 7-UA_/--
Figure A15. Normalized Solid Temperature at _" = 20.0
49
Figure A16. Normalized Solid Temperature at T = 30.0
5O